The higher derivatives of motion are rarely discussed in the teaching of classical mechanics of rigid bodies; nevertheless, we experience the effect not only of acceleration, but also of jerk and snap. In this paper we will discuss the third and higher order derivatives of displacement with respect to time, using the trampolines and theme park roller coasters to illustrate this concept. We will also discuss the effects on the human body of different types of acceleration, jerk, snap and higher derivatives, and how they can be used in physics education to further enhance the learning and thus the understanding of classical mechanics concepts.

This comment addresses the Jiménezet al paper (2008Eur. J. Phys.29 163–75). We correct these authors' expressions for the electric field and the Poynting vector, and also correct their expressions for the magnetic field. Contrary to the authors, we find that the contribution of the second term in Jefimenko's formula for the electric field is not zero. These corrections in no way affect the main points made by the authors of the aforementioned paper, and help strengthen their work.

This work originated as a project in experimental physics conducted by students in the Laboratory of Mechanics and Thermodynamics, a course designed for first-year Physics bachelors. The students were tasked with studying the motion of rigid bodies while having only been introduced to the laws of point-mass dynamics in their theoretical classes. In the proposed experiment, students discovered that bodies with circular symmetry and identical shapes take the same time to roll down an inclined plane. By appropriately fitting the experimental data, they observed that the expression for travel time is equivalent to that of a point mass, except for a multiplicative factor unique to each category of objects. This factor depends on the body's geometry and is directly related to its moment of inertia. Furthermore, we discuss how a similar result can be derived using scaling analysis, illustrating the power of this tool for students early in their physics education. This work may serve as an effective introduction for first-year Physics students to the moment of inertia, as it naturally emerges from a classic physics experiment: the inclined plane.

With the advent of quantum technology, the need for affordable, flexible and robust laboratory experiments not only for students, but also at high school level is increasing. Here, for the first time, we report on a simple modular 3D printed low-cost (<250 €) setup which fulfils these needs for quantum sensing experiments based on nitrogen-vacancy centers in diamonds. Commercially available setups for optically detected magnetic resonance in microdiamonds used as quantum sensor for magnetic fields are not only beyond the reach of any high school (>10 000 €), but also have shortcomings from a didactical point of view, as all the components of the setup are hidden within a 'black box', doomed to be successful 'plug and play'. In contrast, our open-source experimental kit consists of optical components that are placed inside 3D printed open-framed cubes, that can be arranged freely on a grid. This modular and flexible design can provide an inquiry-based learning experience both at undergraduate and high school level.

The paper aims to fulfil three main functions: (1) to serve as an introduction for the physics education community to the functioning of large language models (LLMs), (2) to present a series of illustrative examples demonstrating how prompt-engineering techniques can impact LLMs performance on conceptual physics tasks and (3) to discuss potential implications of the understanding of LLMs and prompt engineering for physics teaching and learning. We first summarise existing research on the performance of a popular LLM-based chatbot (ChatGPT) on physics tasks. We then give a basic account of how LLMs work, illustrate essential features of their functioning, and discuss their strengths and limitations. Equipped with this knowledge, we discuss some challenges with generating useful output withChatGPT-4 in the context of introductory physics, paying special attention to conceptual questions and problems. We then provide a condensed overview of relevant literature on prompt engineering and demonstrate through illustrative examples how selected prompt-engineering techniques can be employed to improveChatGPT-4's output on conceptual introductory physics problems. Qualitatively studying these examples provides additional insights into ChatGPT's functioning and its utility in physics problem-solving. Finally, we consider how insights from the paper can inform the use of LLMs in the teaching and learning of physics.

In both Upper Secondary School and first-level university courses, Newtonian gravitation is typically taught with a primary focus on theoretical aspects, particularly on solving the two-body problem. The solution is claimed reproduce Kepler's empirical laws, but little attention is given to the details of its validation. In the past, the emphasis on theory could be justified by the technical difficulties associated with accessing and analyzing data. Today, however, the availability of open-access data and advanced computational tools makes it feasible to construct empirical evidence for Newtonian gravitation, even within classroom settings. This work revisits the arguments Newton himself used to support his theory and replicate the validation process through a series of activities designed for educational purposes. This approach includes analyzing astrophotographic and mareographic data, along with revisiting Aristarchus of Samos's method for measuring the distance to the Moon through eclipse analysis.

This article deals with the context of seismology and its relevance to the teaching of physics in a modern and highly motivating way. A modular system called Raspberry Shake RS1D is presented, which is capable of detecting earthquakes as well as everyday vibrations. The Raspberry Shake 1D Vertical Motion Seismograph combines a Raspberry Pi mini-computer, a vertical geophone, a 24-bit digitizer and a near-real-time (miniSEED) data transmission. While the Raspberry Shake is a low cost professional and already well known tool for seismologists, we as physicists 'simply' used that computer-based measurement system. We describe how the seismograph works. We present the detection of ground motion from earthquakes as well as from everyday vibrations using selected examples. Thereby, we focus on teaching physics in the context of seismology, presenting didactic ideas for learning about acoustics, for utilizing measurement systems and for applying scientific methods. This article is especially relevant for teaching undergraduate students. Basic facts about earthquakes that are essential for the study of the seismic phenomena discussed in this article are conveyed.

In this comment, some remarks are proposed on José-Philippe Pérez's article 'Redshift formulas and the Doppler–Fizeau effect' (2016Eur. J. Phys.37 015604). Presentation of the gravitational redshift in second part of section 3.1 of that article is shown to be incorrect. The correct derivation is presented.

Problem-solving in physics and mathematics has been characterised in terms of five phases by Schoenfeld, and these have previously been used to describe also online and blended behaviour. We argue that expanding the use of server logs to make detailed categorisations of student actions can help increase knowledge about how students solve problems. We present a novel approach for analysing server logs that relies on network analysis and principal component analysis. We used the approach to analyse student interactions with an online textbook that features physics problems. We find five 'components of behavioural structure': Complexity, Linear Length, Navigation, Mutuality, and Erraticism. Further, we find that problem-solving sessions can be divided into four overarching groups that differ in their Complexity and further into ten clusters that also differ on the other components. Analysing typical sessions in each cluster, we find ten different behavioural structures which we describe in terms of Schoenfeld's phases. We suggest that further research integrates this approach with other methodological approaches to get a fuller picture of how learning strategies are used by students in settings with online features.

A simple analytic model for climate and climate change suitable for physics education in upper secondary and tertiary education is presented. It falls into the class of zero-dimensional energy balance models, comprises five independent variables and uses mathematical techniques only at a moderate level. It allows for the definition and discussion of key concepts of climate science such as radiative forcing, climate sensitivity and feedback factors, including an elementary feedback analysis for the four most important fast climate feedbacks. To round off the text, it is briefly shown how the model can be coupled to the ocean as a heat reservoir, providing a rough estimate for the characteristic time scales of climate change. The analysis is formulated rigorously in terms of the model variables.

We reconsider the dynamics of a point charge in the field of a point-like electric dipole. The force is noncentral, so the angular momentum is not conserved. Our discussion is largely based on the third conserved quantityβ (in addition to the total energy and the axial component of angular momentum) that we derive in spherical coordinates. The equations of motion admit orbits characterised byβ = 0 which are traced on a sphere centred at the dipole. Transitions between orbits on the same sphere demand highly specialised, restricted perturbations of the angular momentum components. If the restrictions are not satisfied, thenr(t) eventually goes to zero or becomes unbounded. This is due to the lack of a confining, radial effective potential. Transitions between orbits lying on a sphere are exceedingly unlikely under realistic conditions. The stability properties of this classical system are virtually unobservable and inconsequential from a physical point of view. Our approach clarifies and extends previous discussions of this matter, and should be accessible to intermediate or advanced undergraduate physics students.

How far can we see with the naked eye at night? Many celestial objects like stars and galaxies as well as transient phenomena such as comets and supernovae can be observed in the night sky. We discuss the furthest distances of such objects and phenomena observable with the naked eye during the night-time for Earth-bound observers. The physics of night-time visual ranges differs from that of daytime observations because human vision shifts from cones to rods. In addition, mostly point sources are observed due to the large distances involved. Whether celestial objects and phenomena can be detected depends on the contrast of their radiation and the background sky luminance. We present a concise overview of how far we can see at night by first discussing the effects of the Earth's atmosphere. This includes attenuation of transmitted radiation as well as its role as a source of background radiation. Disregarding the attenuation of light due to interstellar and intergalactic dust, simple maximum night-time visual range estimates are based on the inverse square law, which can be easily verified by laboratory and demonstration experiments. From the respective calculations, we find that individual stars within the Milky Way galaxy of up to 15 000 light years are observable. Even further away are observable galaxies with several billion stars. The Andromeda galaxy can be observed with the naked eye at a distance of around 2.5 million light years. Similarly, the observability of supernovae also allows a visual range beyond the Milky Way galaxy. Finally, gamma ray bursts as the most energetic events in the universe are discussed concerning naked eye observations.

We discuss the farthest objects on Earth observable for the unaided, healthy naked eye during the daytime, i.e., the maximum visual range for observers on Earth. Visual range depends first on the properties of the material between observer and object and its interaction processes with radiation, but second also on our visual perception system. After a rough comparison of ranges in water, glass, and the atmosphere, we focus on the physical basis of visual range for the latter. As a contrast phenomenon, visual range refers to allowed light paths within the atmosphere. It results from the interplay of geometry, refraction, and light scattering. We present a concise overview of this field by qualitative descriptions and quantitative estimates as well as classroom demonstration experiments. The starting point is the common geometrical visual ranges, followed by extensions due to refraction and limitations due to contrast, which depend on scattering and absorption processes within the atmosphere. The quantitative discussion of scattering is very helpful to easily understand the huge ranges in nature from meters in dense fog to hundreds of kilometers in clear atmospheres. Extreme visual ranges from about 300 km to above 500 km require optimal atmospheric conditions, cleverly chosen locations and times, and a sophisticated topography analysis. Even longer visual ranges are possible when looking through the vertical atmosphere. From the ISS, daytime ranges well above 1000 km are possible.

In this paper, we present an approach to learning about statistical distributions of quantum particles suitable for Advanced Placement high school courses and early-year undergraduate physics and chemistry courses. The procedure developed here uses selected tools known from phenomenological thermodynamics, particularly the concepts of chemical potential and chemical equilibrium, which allow us to circumvent the extensive mathematical apparatus traditionally employed for obtaining equilibrium conditions. To this end, we (a) introduce the notion ofsite, i.e. an abstract 'place' that is able to accept particles; (b) assume that sites having different occupation numbers can be treated as differentelementary substances; (c) consider the change of occupation number as a reaction between elementary substances; and (d) derive quantum distributions by assuming that at equilibrium the driving force, i.e. the difference of chemical potentials of 'educts' and 'products' vanishes. Assuming that the available sites can be occupied either by at most one particle or by any number of them, we obtain the Fermi–Dirac and the Bose–Einstein distributions, respectively. We illustrate a number of examples, including the Planck distribution for the blackbody radiation and the pressure of the degenerate electron gas in white dwarfs. The paper ends with a comparison of quantum and classical Boltzmann distributions and with some remarks from an educational perspective.

A simple method to obtain the general formula for the Thomas precession rate is provided. The approach enables us to grasp the effect of the Thomas precession in a visual and intuitive manner.

It is universally believed that with his 1905 paper 'Does the inertia of a body depend on its energy content?' Einstein first demonstrated the equivalence of mass and energy by making use of his special theory of relativity. In the final step of that paper, however, Einstein equates the kinetic energy of a body to its Newtonian value, indicating that his result is at best a low-velocity approximation. Today, several characters debate whether a mid-nineteenth century physicist, employing only physics available at the time, could plausibly arrive at the celebrated result. In other words, is Einsteinian relativity necessary to derive?

Although synchronization effects play an important role in many areas of basic and applied science, their treatment in undergraduate physics courses requires more attention. Based on acoustic experiments with a driven organ pipe, the article proposes analytical, numerical and qualitative approaches to this universal phenomenon, suitable for introductory teaching. The Adler equation is developed, a first-order nonlinear differential equation describing the phase dynamics of driven self-sustained oscillations in the weak coupling limit. Analytical solutions, intuitive mechanical analogues and properties of the resulting comb spectra are discussed. The underlying phase model is paradigmatic for synchronization-based self-organization phenomena in a wide range of fields, from physics and engineering to life and social sciences.

The use of augmented reality (AR) allows for the integration of digital information onto our perception of the physical world. In this article, we present a comprehensive review of previously published literature on the implementation of AR in physics education, at the school and the university level. Our review includes an analysis of 96 papers from the Scopus and Eric databases, all of which were published between 1st January 2012 and 1st January 2023. We evaluated how AR has been used for facilitating learning about physics. Potential AR-based learning activities for different physics topics have been summarized and opportunities, as well as challenges associated with AR-based learning of physics have been reported. It has been shown that AR technologies may facilitate physics learning by providing complementary visualizations, optimizing cognitive load, allowing for haptic learning, reducing task completion time and promoting collaborative inquiry. The potential disadvantages of using AR in physics teaching are mainly related to the shortcomings of software and hardware technologies (e.g. camera freeze, visualization delay) and extraneous cognitive load (e.g. paying more attention to secondary details than to constructing target knowledge).

Language is an important resource for physicists and learners of physics to construe physical phenomena and processes, and communicate ideas. Moreover, any physics-related instructional setting is inherently language-bound, and physics literacy is fundamentally related to comprehending and producing both physics-specific and general language. Consequently, characterizing physics language and understanding language use in physics are important goals for research on physics learning and instructional design. Qualitative physics education research offers a variety of insights into the characteristics of language and language use in physics such as the differences between everyday language and scientific language, or metaphors used to convey concepts. However, qualitative language analysis fails to capture distributional (i.e. quantitative) aspects of language use and is resource-intensive to apply in practice. Integrating quantitative and qualitative language analysis in physics education research might be enhanced by recently advanced artificial intelligence-based technologies such as large language models, as these models were found to be capable to systematically process and analyse language data. Large language models offer new potentials in some language-related tasks in physics education research and instruction, yet they are constrained in various ways. In this scoping review, we seek to demonstrate the multifaceted nature of language and language use in physics and answer the question what potentials and limitations artificial intelligence-based methods such as large language models can have in physics education research and instruction on language and language use.

This pedagogical article elucidates the fundamentals of trapped-ion quantum computing, which is one of the potential platforms for constructing a scalable quantum computer. The evaluation of a trapped-ion system's viability for quantum computing is conducted in accordance with DiVincenzo's criteria.

Motivated by filling the gap we felt after years of teaching analytical mechanics, a non relativistic, classical introduction to Lagrangian mechanics has accordingly been provided here, which covers all possible forms of Euler-Lagrange equation, derived through dealing with different kinds of forces including conservative forces, forces of constraint, velocity-dependent forces, and non-conservative/dissipative forces. Hamiltonian mechanics has also been concluded as a reformulation of Lagrangian mechanics via applying Legendre transformation. Ignorable coordinates have finally been introduced, leading to Hamilton-Jacobi formalism, from which an&#xD;equivalence between dynamics of a classical point particle and that of a plane wave has been inferred. We have showed that such an equivalence had long laid the required theoretical ground for the advent of wave mechanics; therefore, a number of landmark advancements in theoretical physics, including Hamiltonian mechanics, canonical transformations, and formulations of quantum mechanics have roots in Lagrangian mechanics.

In this article, we present an experimental pedagogical proposal for the study of material strength. Our objective is to experimentally obtain deflection curves by proposing the analysis of images using the free software GeoGebra as a method for obtaining experimental data. We will apply our methodology to the study of a beam under two deflection conditions. First, we analyze a cantilever beam under the action of a transverse load that causes bending, and then we apply the proposed technique to the case of a beam subjected to two opposing forces that cause buckling. The proposal of this article is to design portable and low-cost experiments. As a result, the&#xD;experimental values of the bending curve for different loadings are obtained. These curves were compared with the bending theory, allowing the determination of the Young's modulus of the cantilever beam material E = (64.3 ± 1.3) GPa. The value of the critical axial force required to cause buckling instability was estimated Fcr = 2.3N . Our methodology shows a fit of the experimental results with the theoretical predictions with R2 ≥ 0.9 and good agreement with literature.

We present educational material about Bell inequalities in the context of quantum computing.&#xD;In particular, we provide software tools to simulate their violation, together with a&#xD;guide for the classroom discussion. The material is organized in three modules of increasing&#xD;difficulty, and the relative implementation has been written in Qibo, an open-source software&#xD;suite to simulate quantum circuits with the ability to interface with quantum hardware.&#xD;The topic of inequalities allows not only to introduce undergraduate or graduate students&#xD;to crucial theoretical issues in quantum mechanics – like entanglement, correlations, hidden&#xD;variables, non-locality –, but also to practically put hands on tools to implement a real simulation,&#xD;where statistical aspects and noise coming from current quantum chips also come&#xD;into play.&#xD;

Mirror astigmatism is analysed with different approaches in the two orthogonal planes, in a way that is easier to follow and understand than the classic textbooks usually referred to in research papers.&#xD;

The Bernoulli levitation demonstration using a funnel and a ball is widely used to teach the Bernoulli principle. But the claim is made in the literature that there are other effects than the Bernoulli principle, and that solving the problem requires numerical fluid physics calculations. We measure the upward-acting force in such an experiment for a selection of balls with radii from 19.67 mm to 39.07 mm, and with weights from 0.037 N to 0.159 N. We find that, when the ratio of the radius of the ball to radius of the inner opening of the funnel is greater than 2, and the ratio of ball radius is less than the wider opening of the funnel, the Bernoulli effect dominates. &#xD;

How far can we see with the naked eye at night? Many celestial objects like stars and galaxies as well as transient phenomena such as comets and supernovae can be observed in the night sky. We discuss the furthest distances of such objects and phenomena observable with the naked eye during the night-time for Earth-bound observers. The physics of night-time visual ranges differs from that of daytime observations because human vision shifts from cones to rods. In addition, mostly point sources are observed due to the large distances involved. Whether celestial objects and phenomena can be detected depends on the contrast of their radiation and the background sky luminance. We present a concise overview of how far we can see at night by first discussing the effects of the Earth's atmosphere. This includes attenuation of transmitted radiation as well as its role as a source of background radiation. Disregarding the attenuation of light due to interstellar and intergalactic dust, simple maximum night-time visual range estimates are based on the inverse square law, which can be easily verified by laboratory and demonstration experiments. From the respective calculations, we find that individual stars within the Milky Way galaxy of up to 15 000 light years are observable. Even further away are observable galaxies with several billion stars. The Andromeda galaxy can be observed with the naked eye at a distance of around 2.5 million light years. Similarly, the observability of supernovae also allows a visual range beyond the Milky Way galaxy. Finally, gamma ray bursts as the most energetic events in the universe are discussed concerning naked eye observations.

In this paper, we present an approach to learning about statistical distributions of quantum particles suitable for Advanced Placement high school courses and early-year undergraduate physics and chemistry courses. The procedure developed here uses selected tools known from phenomenological thermodynamics, particularly the concepts of chemical potential and chemical equilibrium, which allow us to circumvent the extensive mathematical apparatus traditionally employed for obtaining equilibrium conditions. To this end, we (a) introduce the notion ofsite, i.e. an abstract 'place' that is able to accept particles; (b) assume that sites having different occupation numbers can be treated as differentelementary substances; (c) consider the change of occupation number as a reaction between elementary substances; and (d) derive quantum distributions by assuming that at equilibrium the driving force, i.e. the difference of chemical potentials of 'educts' and 'products' vanishes. Assuming that the available sites can be occupied either by at most one particle or by any number of them, we obtain the Fermi–Dirac and the Bose–Einstein distributions, respectively. We illustrate a number of examples, including the Planck distribution for the blackbody radiation and the pressure of the degenerate electron gas in white dwarfs. The paper ends with a comparison of quantum and classical Boltzmann distributions and with some remarks from an educational perspective.

We present educational material about Bell inequalities in the context of quantum computing.&#xD;In particular, we provide software tools to simulate their violation, together with a&#xD;guide for the classroom discussion. The material is organized in three modules of increasing&#xD;difficulty, and the relative implementation has been written in Qibo, an open-source software&#xD;suite to simulate quantum circuits with the ability to interface with quantum hardware.&#xD;The topic of inequalities allows not only to introduce undergraduate or graduate students&#xD;to crucial theoretical issues in quantum mechanics – like entanglement, correlations, hidden&#xD;variables, non-locality –, but also to practically put hands on tools to implement a real simulation,&#xD;where statistical aspects and noise coming from current quantum chips also come&#xD;into play.&#xD;

The quintessential example of a fluid flow problem with a known solution is the drag force required to move a sphere through a stationary fluid. While the equation describing this Stokes drag is simple, deriving it from the Stokes equations requires several pages of mathematics. In this paper, I present an alternative, more intuitive approach, based on the Oseen tensor, which gives the fluid flow due to an applied point force. For completeness, we first derive the expression for the Oseen tensor, then use it to re-derive Stokes law. As an additional application, we also study fluid flow near a wall, using a mirror images technique similar to the one used in electrostatics problems.

The 2019 revision of the International System of Units (SI) set exact values for four defining natural constants, allowing all base and derived units to be defined exactly. While the distinction between base and derived units remains valuable, particularly in education where it helps students grasp the SI's structure more easily, we argue for clarifying the classification of these base units. Currently, the SI defines seven base units, but these fall into two fundamentally different subsets. The first subset—the kilogram (kg), meter (m), second (s), and ampere (A)—forms a foundational set of physics units from which all other units can be derived. The second subset—the candela (cd), kelvin (K), and mole (mol)—serves a different role. We propose recognizing this difference by classifying the candela, kelvin, and mole as a distinct category, possibly called 'scale-spanning units'. While such a revision would not affect the accuracy of measurement, it would help students and newcomers to metrology to attain better clarity of understanding of the International System of Units.

Mirror astigmatism is analysed with different approaches in the two orthogonal planes, in a way that is easier to follow and understand than the classic textbooks usually referred to in research papers.&#xD;

The present study explored the role of first-year physics students' sense of belonging to physics and how it relates to their learning progression in higher education-level physics. Conducted at Paderborn University, this study examined how students' sense of belonging to physics influences their academic outcomes in an introductory experimental physics course. Additionally, we investigated how physics students' engagement in the Physiktreff—a holistic support program for first-year physics students to help them cope with academic and social challenges during their studies—impacts the development of their sense of belonging to physics over time. Our findings indicated that students with a stronger sense of belonging to physics performed better academically. Moreover, students who actively participated in the support program experienced a positive shift in their sense of belonging to physics. However, our findings also revealed that physics students with a higher initial sense of belonging to physics tended to experience a decline in their sense of belonging to physics during their first semester. These results underscore the importance of fostering a sense of belonging to physics within higher education, particularly during the introductory phase of students' studies.&#xD;

An undergraduate physics experiment is described that uses a Fabry–Perot etalon and digital camera to determine the hyperfine frequency shifts in naturally occurring mercury. Radiation at 546 nm emitted from the 7³S₁ state relaxing to the 6³P₂ state in a low pressure Hg lamp is selected by an interference filter to pass through the etalon. This produces a series of concentric rings at the focus of the camera lens that are analyzed using bespoke software written for this experiment. Results from the analysis are compared to theoretical calculations of the hyperfine shifts for both the¹⁹⁹Hg and²⁰¹Hg isotopes.

This work originated as a project in experimental physics conducted by students in the Laboratory of Mechanics and Thermodynamics, a course designed for first-year Physics bachelors. The students were tasked with studying the motion of rigid bodies while having only been introduced to the laws of point-mass dynamics in their theoretical classes. In the proposed experiment, students discovered that bodies with circular symmetry and identical shapes take the same time to roll down an inclined plane. By appropriately fitting the experimental data, they observed that the expression for travel time is equivalent to that of a point mass, except for a multiplicative factor unique to each category of objects. This factor depends on the body's geometry and is directly related to its moment of inertia. Furthermore, we discuss how a similar result can be derived using scaling analysis, illustrating the power of this tool for students early in their physics education. This work may serve as an effective introduction for first-year Physics students to the moment of inertia, as it naturally emerges from a classic physics experiment: the inclined plane.

The theory of the relativistic Doppler effect is well established and validated by several experiments. Most derivations, however, assume that both the reference frame of the source and the reference frame of the receiver are inertial, and the validity of the obtained formulas in the case of a non-inertial source and/or receiver is usually not clarified in higher physics education. As shown in this paper, if such formulas are carelessly applied to non-inertial cases, they can lead to contradictory results. This is probably what happened historically in the interpretation of experiments with a receiver in uniform circular motion around an inertial source, where some authors reported a shift towards the red, some others a shift towards the blue. This confusion points out the importance of establishing Doppler effect formulas that are valid in non-inertial cases. In this paper, we first set a rigorous definition of the relativistic Doppler shift, clarifying that it is the ratio between two proper time intervals relating to two different pairs of events. We then apply the definition to derive the relativistic Doppler shift in the case of a non-inertial source and/or receiver. The proposed derivations are very simple, provide important insights on the correct application of the time dilatation formula to accelerated systems, and highlight possible subtle mistakes that can lead to completely wrong results. This makes them a useful tool for teaching purposes, especially for graduate students.